Let F be a field and let K be an algebraically closed field with F â?? K.
If f â??F[x] is irreducible (i.e. if f = m * n , then one of m or n is a constant) and f
has a multiple zero in K ,
then f â?² = 0© BrainMass Inc. brainmass.com December 16, 2022, 7:41 am ad1c9bdddf
Proposition. Let F be a field and let K be an algebraically closed field with F⊆K. If f∈F[x] is irreducible and f has a multiple zero in K, then f^'=0.
Proof: Before we prove this proposition, let's introduce a theorem related to separable polynomials and their derivatives. A polynomial f∈F[x] is said to be separable if it has distinct zeros in a splitting field over F. That is, each zero of f has multiplicity 1. We now introduce the following theorem.
Theorem 1. A nonzero polynomial f in F[x] is separable if and only if it's relatively ...
Proposition is clarified here for an algebraically closed field.